An Introduction to Differentiable Manifolds and Riemannian GeometryAcademic Press, 1986/04/21 - 429 ページ An Introduction to Differentiable Manifolds and Riemannian Geometry |
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... mapping F from an open subset U of R" into R*. Here the Jacobian is defined and the mean value theorem restated for mappings. Sections 3 and 4 deal with the concept of the space of tangent vectors T.(R") at a point a e R"; this will be ...
... mapping F from an open subset U of R" into R*. Here the Jacobian is defined and the mean value theorem restated for mappings. Sections 3 and 4 deal with the concept of the space of tangent vectors T.(R") at a point a e R"; this will be ...
22 ページ
... f is differentiable at a, then it is continuous at a and all the partial derivatives (Öfföx'), exist. Moreover the b ... mapping of an open interval (a, b) = {xe R a < x < b) of the real numbers into R", f: (a,b) → R", with f(t) = {x'(t) ...
... f is differentiable at a, then it is continuous at a and all the partial derivatives (Öfföx'), exist. Moreover the b ... mapping of an open interval (a, b) = {xe R a < x < b) of the real numbers into R", f: (a,b) → R", with f(t) = {x'(t) ...
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... f(0) = 0, f(t) = exp(-1/t”) for t + 0; fis C* on R. Is it C" on R2) 7. Prove (1.4), that is, prove that g o f is ... mapping defined on A c R", then F is determined by its coordinate functions f = to o F; in fact for xe A, F(x) = (f'(x) ...
... f(0) = 0, f(t) = exp(-1/t”) for t + 0; fis C* on R. Is it C" on R2) 7. Prove (1.4), that is, prove that g o f is ... mapping defined on A c R", then F is determined by its coordinate functions f = to o F; in fact for xe A, F(x) = (f'(x) ...
26 ページ
... mapping F a smooth mapping; if F is smooth, then each coordinate function f" possesses continuous partial derivatives of all orders and each such derivative is independent of the order of differentiation. If F is differentiable on U, we ...
... mapping F a smooth mapping; if F is smooth, then each coordinate function f" possesses continuous partial derivatives of all orders and each such derivative is independent of the order of differentiation. If F is differentiable on U, we ...
27 ページ
... F(a)|| < (nm)” K|x – al. We will use DF to denote the Jacobian matrix of a differentiable mapping F and DF(x) to denote its value at x. If F is differentiable on U, then for a e U expression (*) becomes F(x) = F(a) + DF(a)(x - a) + |x ...
... F(a)|| < (nm)” K|x – al. We will use DF to denote the Jacobian matrix of a differentiable mapping F and DF(x) to denote its value at x. If F is differentiable on U, then for a e U expression (*) becomes F(x) = F(a) + DF(a)(x - a) + |x ...
目次
1 | |
20 | |
51 | |
Chapter IV Vector Fields on a Manifold | 106 |
Chapter V Tensors and Tensor Fields on Manifolds | 176 |
Chapter VI Integration on Manifolds | 229 |
Chapter VII Differentiation on Riemannian Manifolds | 297 |
Chapter VIII Curvature | 365 |
References | 417 |
Index | 423 |
多く使われている語句
algebra basis bi-invariant C*-vector field compact completes the proof components connected coordinate frames coordinate neighborhood Corollary corresponding countable covariant tensor covering curve p(t defined definition denote derivative diffeomorphism differentiable manifold dimension domain of integration element equations equivalent Euclidean space example Exercise exists fact finite fixed point formula functions geodesic geometry given Gl(n hence homeomorphism homotopy identity imbedding inner product integral curve isometry isomorphism Lemma Let F Lie group G linear map mapping F matrix notation obtain one-parameter subgroup one-to-one open set open subset oriented orthogonal orthonormal parameter plane properly discontinuously properties prove rank real numbers regular submanifold Remark Riemannian manifold Riemannian metric Section Show structure submanifold subspace suppose surface symmetric tangent space tangent vector tensor field Theorem Let topology uniquely determined vector field vector space zero